(PDF) Robust optimization model for sust

The current issue and full text archive of this journal is available on Emerald Insight at: https://www.emerald.com/insight/1746-5664.htm Robust Robust optimization model for optimization sustainable supply chain for model production and distribution of polyethylene pipe Jaber Valizadeh, Ehsan Sadeh and Zainolabedin Amini Sabegh Received 22 June 2019 Revised 26 January 2020 Department of Management, Saveh Branch, Islamic Azad University, Accepted 28 March 2020 Saveh, Iran, and Ashkan Hafezalkotob Industrial Engineering College, South Tehran Branch, Islamic Azad University, Tehran, Iran Abstract Purpose – In this study, the authors consider the key decisions in the design of the green closed-loop supply chain (CSLC) network. These decisions include considering the optimal location of suppliers, production facilities, distribution, customers, recycling centers and disposal of non-recyclable goods. In the proposed model, the level of technology used in recycling and production centers is taken into account. Moreover, in this paper is the environmental impacts of production and distribution of products based on the eco-indicator 99 are considered. Design/methodology/approach – In this study, the author consider the key decisions in the design of the green CLSC network. These decisions include considering the optimal location of suppliers, production facilities, distribution, customers, recycling centers and disposal of non-recyclable goods. In the proposed model, the level of technology used in recycling and production centers is taken into account. Moreover, the environmental impacts of production and distribution of products based on the eco-indicator 99 are considered. Findings – The results indicate that the results obtained from the colonial competition algorithm have higher quality than the genetic algorithm. This quality of results includes relative percentage deviation and computational time of the algorithm and it is shown that the computational time of the colonial competition algorithm is signiﬁcantly lower than the computational time of the genetic algorithm. Furthermore, the limit test and sensitivity analysis results show that the proposed model has sufﬁcient accuracy. Originality/value – Solid modeling of the green supply chain of the closed loop using the solid optimized method by Bertsimas and Sim. Development of models that considered environmental impacts to the closed loop supply chain. Considering the impact of the technology type in the manufacture of products and the recycling of waste that will reduce emissions of environmental pollutants. Another innovation of the model is the multi-cycle modeling of the closed loop of supply chain by considering the uncertainty and the ﬁxed and variable cost of transport. Keywords Algorithms, Optimization, Manufacturing, Mathematical programming, Modelling, Supply chain management, Closed loop supply chain, Polyethylene pipe, Sustainable, Robust optimization Paper type Research paper Journal of Modelling in Management © Emerald Publishing Limited 1746-5664 Islamic Azad University Central Tehran Branch. DOI 10.1108/JM2-06-2019-0139 JM2 1. Introduction The consultants originally introduced the term of supply chain management (SCM) in the early 1980s and subsequently so much attention has been paid to it Lambert and Cooper (2000). Designing the supply chain network includes all internal and external components of SCM (Tiwari et al., 2016). The ﬁrst aim of a chain is to supply and satisfy the customer needs in the process of producing value for itself. Supply chain activities start according to the customer’s order and end when the customer pays for the goods and the received services. Most chains are actually networked (Christopher and Gattorna, 2005). The green supply chain is a type of supply chain in which environmental initiatives have been taken into account (Soleimani et al., 2017). Green supply chain leads to improve environmental conditions, economic performance and competitiveness (Khalilzadeh and Derikvand, 2018). To curb the ever increasing degradation of environment, fast deteriorating supply of natural minerals and resources and proper waste disposal have made governments act upon these concerns globally (Prakash et al., 2017). If both demand and supply chains are considered simultaneously, the resulting network is deﬁned as a closed-loop supply chain (CLSC) (Govindan et al., 2015). The CLSC management, which is one of the most signiﬁcant management issues, has been increasingly spotlighted by the government, companies and customers, over the past years (Ghahremani-Nahr et al., 2019). CLSC focuses on recycling the products according to the customers (Faccio et al., 2014). Reverse logistics and CLSCs have become the front line of planning in recent years by researchers and decision-makers because of the government regulations, increased attention to environmental impacts and conservation of natural resources (Soleimani et al., 2017). On the other hand, distribution of CLSC network is one of the most important issues in SCM (Tiwari et al., 2016). A CLSC needs designing, controlling and running a system to maximize the value creation over the life of a product by generating a dynamic value of different returned products over the time (Govindan et al., 2015). Designing and utilization of the supply chain network is a strategic decision that its effects continue for several years and commercial space parameters such as customers demand may change. So, it can be considered that these parameters have a non- deterministic nature and the designed supply chain should be consistent against such uncertainties (Meepetchdee and Shah, 2007). A stable optimization is an optimization technique that tries to provide a stable model and a solution in a complete uncertainty space according to two scenario-based and deterrent approaches. The stability of the model, the feasibility and the stability of the answer guarantee the optimality of the created responses against the appearance of different states of the parameter with uncertainty (Pan and Nagi, 2010). Due to the long-term planning horizon in designing a supply chain, the assumption of some parameters 0 certainty is a bit far from reality. Factors such as demand, supply of raw materials, the price of raw materials and ﬁnal products and the markets location have uncertainties in the supply chain life cycle (Schutz et al., 2009). On the other hand, moving towards global markets and creating transnational unions such as the European Union has brought the environment and inﬂuenced factors on decision-making with new uncertainties items such as exchange rate and reliability of distribution channels (Santoso et al., 2005). Hence, we need a coherent joint chain that exhibits minimal impact against the environmental changes. In supply chain modeling, there are always parameters that cannot be accurately estimated once they are realized in the future. However, we can consider the range of variation and even the probability distribution for the values that these parameters can take. Undoubtedly, taking into account this information is better than average estimates of supply chain modeling. One of these methods of dealing with uncertainty that has been widely used in real-world problems is the optimization method. In this research, we have Robust used Bertsimas and Sim based robust optimization method, which are the most applicable optimization and efﬁcient methods of robust optimization in real-world problems. It has been attempted to use a robust optimization technique in this research to prepare a model to design a multi- model product, single-cycle and multi-level supply chain and considering the environmental labels such as a sustainable green supply chain. The robust optimization prepares the possibility to minimize the effects from the parameters associated with the affective uncertainties in the solutions of the model. The main issue of a logistics management problem is contrasting the uncertainty in the future. Most logistic models consider the possibility of uncertainty in their variables or parameters based on their experience. They also consider the environmental investment. Wang and Hsu (2010), were prepared one of the ﬁrst research studies to investigate the environmental investment. On the other hand, environmental effects of the production, transporting, recycling and disposal of polyethylene pipes have been used using the eco-indicator 99, which it is used by Goedkoop and Spriensma (2000). Furthermore, multi-cycle modeling in CLSC has been done regarding the uncertainty and considering the ﬁxed and variable cost of transportation. There are many virtues for recycling various types of polymers such as saving costs. We all know that polymer materials are decomposed during a millennium in the environment, and this non-degradability has raised the idea of recycling. The recycling of all types of polyethylene pipes makes it possible to use products, which are useful in the textile, automotive, automotive and other industries. The usage of these materials with the same quality as the ﬁrst production has absorbed more researchers’ attention to the recycling problem. Substances such as polyethylene pipes should be recycled properly and used in the environment several times that leads to reduction of production costs. In this study, we show how decisions on reverse logistics help meet environmental requirements. Therefore, we model the CLSC processes using the product life cycle assessment method and consider the total cost of the supply chain by considering the environmental requirements as the performance benchmark of this supply chain. This model ensures that in addition to reducing total costs, environmental requirements are met. Besides, given the rate of return on products, the proposed model gets closer to the real world. In general, in this article, we scrutinize how to provide a solid model for the green supply chain and closed loop for the production and distribution of polyethylene pipes. The purpose of organizations and companies is to maintain and increase proﬁts as well as to survive in the marketplace because the globalization of economic activities coupled with the rapid growth of technology and limited resources has put companies in close competition. One of the competitive advantages is to make operations such as the supply chain more efﬁcient and effective. Furthermore, due to government regulations, environmental issues and the development of the concept of social responsibility, CLSC management has been the subject of much interest. Therefore, as recycling plays an important role in the sustainability of the earth, human life and other organisms, if reusable goods are not reused, in addition to heavy costs for companies they will have adverse effects on the supply chain. Therefore, it is necessary to study these conditions. Table 1 summarizes the research gap in this area and the features and innovations of this research. In the following, to clarify the issue, the issue of the study and its associated assumptions are discussed in detail. In particular, this is an attempt to answer the following research queries: RQ1. How does this approach end up with saving on supply chain costs by modeling the activities of recycling, producing and distributing polyethylene pipes? JM2 related research Examination of Table 1. Certainty Network Period numbers Environmental Multi- Single- Researcher criterion Uncertain Certain Reverse Straight period period Solution Nagurney and Toyasaki (2005) Weighting method Melo et al. (2009) Lagrangian liberation Salema et al. (2007) Branch and border Lieckens and Vandaele (2007) Innovative (queuing techniques) Listes (2007) Branch and cut Thanh et al. (2008) Branch and border Chouinard et al. (2008) SAAþ Cplex 9 Wang and Hsu (2010) CPLEX Yeh and Chuang (2011) MOGA Algorithm Tseng et al. (2014) Fuzzy ANP method using Fahimnia et al. (2015) CPLEX Rezaee et al. (2015) Accurate resolution using CPLEX Huang et al. (2016) Genetic Algorithm Tiwari et al. (2016) NSGA II Golpîra et al. (2017) The Karush-Kuhn- Tucker Hajiaghaei-Keshteli and Fathollahi Multi algorithm Fard (2018) Gholizadeh et al. (2018) Genetic algorithm Zhen et al. (2019) The exact solution method Ghahremani-Nahr et al. (2019) Whale optimization algorithm Zhao and Sun (2019) Game theory Lejarza and Baldea (2020) The exact solution method Santander et al. (2020) CPLEX The present research (2020) Bertsimas and Sim stable optimization approach Robust RQ2. How to provide a comprehensive approach for companies to meet environmental optimization requirements and obtain environmental permits in the organization’s business model environment? RQ3. How to calculate environmental labels of recycling, production and distribution of polyethylene pipes through supply chain modeling? The purpose of this study is to build a model to design a multi-product supply chain, single- cycle, multi-level using a robust optimization technique and along with the consideration of environmental labels. In other words, a sustainable green supply chain would be provided. The robust optimization makes the responses produced by the model have the least impact from the parameters associated with the uncertainties affecting decision-making. The main issue is the logistics management problem, when it comes to the uncertainty of the future. Most logistic models, based on their experience are aware of the possibility of uncertainty in variables or their parameters. This modeling has been carried out with Bertsimas and Sim (2004) optimization method. On the other hand, the indices of eco-indicator 99 were used to examine the environmental effects of the production, transport, recycling and disposal of polyethylene pipes used by Goedkoop and Spriensma (2000). The reminder letter is as follows. The review of the literature review is in Section 2. The model formulation is presented in Section 3. Section 4 provides a numerical example and a desirable solution to reduce costs in the supply chain of polyethylene pipes. Moreover, analytical results and sensitivity analysis are discussed. Final remarks and some aspects for future research are presented in Section 5. 2. Literature review This research is closely related to close loop supply chain (with collecting returned products) as well sustainable supply chain, which are reviewed through following subsections. 2.1 Closed-loop supply chain Many articles have been published on the CLSC. However, before 2001, many of them involved only reverse logistics provided a network of factories, warehouses and dismantling centers (Fleischmann et al., 1997). In other papers, the importance of the topic was revealed by a case and an example examination in the industry. Take Sheu’s paper, 2005, examining factories for the production of computers in Taiwan. Factors such as the used-product return ratio and corresponding subsidies from governmental organizations for reverse logistics are considered in the model formulation (Sheu et al., 2005) or metal mechanics were studied in 2008. Fuente et al. (2008). The proposed integrated model for supply chain management is aimed at re-deﬁning demand management procedures, order management procedures, manufacturing management procedures, procurement management procedures, distribution management procedures, client management procedures, etc. This trend continued in the published articles afterward. For instance, in the single-core ﬁeld and the deﬁnitive space, Lu and others in 2007, provided a network including manufacturers, the centers of remanufacturing, resettlement and intermediary. In their models, goods are delivered to customers after manufacturing by the manufacturers (Lu and Bostel, 2007). Recycled goods are grouped into two categories after being collected by the intermediary centers. Goods to be ousted and the ones to be delivered to producers for a series of operations. In 2010, Wang and Hsu provided papers in deﬁnitive and monotonous space. The provided network included four layers, namely, factories, distribution centers, suppliers and recycling centers JM2 (Wang and Hsu, 2010). It should be noted that customer demand and the amount of returning goods are not necessarily a certain amount, hence, uncertainty plays an important role in this regard. There are several articles in this area, for example, Melo et al. (2009) can be referenced for the actions both carried out in this ﬁeld and the deﬁnite space’s in the closed loop supply chain using game theory. Yuan et al. (2020) adopted the method of optimization analysis and game theory to analyze the necessary conditions for manufacturers, retailers and third-party online recyclers to achieve interest coordination under equilibrium conditions. 2.2 Sustainable supply chain In 2010, sustainable SCM became the focus of researchers (Pereira de Carvalho and Barbieri, 2012). Sustainable SCM is the ﬂow management of materials, information, capital and collaboration between supply chain companies with the development of three dimensions of sustainability, namely economic, environmental and social dimensions (Ageron et al., 2012). Sustainable product design, process design and sustainable collaboration with suppliers as well as customers are the four main dimensions of sustainable SCM (Paulraj et al., 2017). As stakeholder awareness of environmental and social issues grows, interest in sustainability is increasing among academics and professionals (Mani et al., 2018). The environment has been one of the key elements of the tripartite sustainability policy and an intermediary for issues such as climate change and rising energy prices (Valizadeh and Alizadeh, 2017). To integrate sustainability goals at the supply and organizational level, the initial impetus for company decisions comes from external pressures and incentives and usually from focal companies to their suppliers in the process of sustainability and supply chain concept transfer (Neutzling et al., 2018). Sustainability helps hire people who care about the environmental issues. Sustainable innovation goes beyond the power of a company alone and relates to the supply chain of companies (Gao et al., 2016). Carter and Rogers (2008) have argued that economic transactions can provide insights into why the supply chain is developing. Kamble et al. (2020) studied sustainable performance in a data-driven agriculture supply chain. 2.3 Green supply chain Besides, many researchers have considered the issues of environmental and social responsibility in their studies recently. Govindan et al. ((2015). They selected and reviewed total of 382 papers during January 2007 and March 2013. The papers were analyzed and categorized to construct a useful foundation of past research. De and Giri (2020) studied a CLSC focusing on managing, scheduling and routing problems to achieve economic and environmental sustainability. Zhao and Sun (2019) conducted a distribution analysis of income distribution based on government subsidies in the closed loop supply chain using game theory. Li et al. (2019) developed a comprehensive decision model for sustainable design of coal supply chain. Resat and Unsal (2019) presented a novel two-stage hybrid solution method designed for sustainable supply chain systems in packaging industry. Tseng (2010) has evaluated green supply chain management (GSCM) criteria in selecting the suppliers using the theory of fuzzy sets. Mirhedayatian et al. (2014) have also evaluated the GSCM trough data envelopment analysis (DEA) technique. According to the DEA logic, generating higher outputs with lower inputs is a performance criterion. In a paper by Gavronski et al. (2011), it was observed that the organization’s resources have a positive relationship with chain management capabilities using a resource-based perspective. The data from a survey of a sample of manufacturing plants indicates that a managerial philosophy that includes external knowledge exchange directly supports both greener process management and environmental collaboration with suppliers. Yeh and Chuang (2011) presented a math-planning model for choosing green supply chain partners Robust considering four goals including the cost, time, quality and green assessment. They optimization introduced green criteria into the framework of supplier selection criteria and developed an optimum mathematical planning model for green partner selection, which involved four model objectives such as cost, time, product quality and green appraisal score. To solve these conﬂicting objectives, they adopted two multi-objective genetic algorithms to ﬁnd the set of pareto-optimal solutions, which used the weighted sum approach that can generate more number of solutions. Nagurney and Toyasaki (2005) presented a new model of designing a balanced green supply chain network with uncertain demand by considering the environmental factors. They applied weighting method to solve the proposed model. Tseng et al. (2014) studied the selection of green suppliers using fuzzy ANP. They proposed a combined fuzzy grey relational analysis method based to deal with study objective. Objective’s them were aimed to present a perception approach to deal with supplier evaluation of environmental knowledge management capacities with uncertainty and lack of information. Fahimnia et al. (2015) presented a nonlinear complex integer-programming model for tactical level modeling of green supply chain. This model was studied to create a trade-off between costs and environmental considerations, including carbon emissions, energy and water consumption. Rezaee et al. (2015) studied a two-stage randomized design model to design a green supply chain in an environment that was prepared for credits and markets for carbon emissions by the governments. The annual right for carbon emission is given to the companies, which this issue is called carbon credits. Huang et al. (2016) studied a green supply chain in their current research studies by several suppliers, a manufacturer and several retailers. Then, a game theory model was used to study the effects of designing a production line, selecting suppliers, choosing vehicles and pricing strategies on greenhouse gas emissions and proﬁts, simultaneously. In Table 1, some of research studies are examined more closely. On the other hand, the systematic and environmental uncertainties arise from the long-term planning of the supply chain and the dynamic nature of the industry. Systematic uncertainties include uncertainties in production, distribution, collection and recycling processes. For example, the uncertainty of the delivery time, the production cost and the capacity of various facilities, fall into this category (Pishvaee et al., 2011). One of the common methods to deal with uncertainties in the supply chain is possible planning, which has some problems in making efﬁcient and reliable decisions (Vahdani et al., 2012). El-Sayed et al. (2010) proposed an integrated probabilistic model to design a forwarding and reverse integrated logistic network according to the uncertainty of demand and return rate and the goal was to maximize the overall proﬁts. The proposed network structure consists of three echelons in the forward direction (i.e. suppliers, facilities and distribution centers) and two echelons, in the reverse direction (i.e. disassembly and redistribution centers), ﬁrst customer zones in which the demands are stochastic and second customer zones in which the demand is assumed to be deterministic, but it may also assumed to be stochastic. Dekker et al. (2011), have currently done a review about designing Green Logistics Networks, however, SCM research has also recently included the environmental studies. The primary classiﬁcation of the aims of this research is as follows: Investment decisions related to environmental topics were considered, in this study. 2.4 Research gap and contributions In recent years, some research in the supply chain considered some parameters with uncertainty in their proposed model. Most of these papers beneﬁt from probable planning. The proposed models considered simple assumptions such as mono-product. However, we have extended it to multi-products. Another difference is its green nature so that in the JM2 articles environmental impacts are not considered as production constraints. In this model, owing to the reversal nature of the product, we consider the model multi-cyclical. The aspects of research innovation can be summarized as follows: Solid modeling of the green supply chain of the closed loop using the solid optimized method by Bertsimas and Sim. Development of models that considered environmental impacts to the closed loop supply chain. Goedkoop and Spriensma (2000) and Pishvaee and Razmi (2012) are of the articles examining the environmental impacts. Considering the impact of the technology type in the manufacture of products and the recycling of waste that will reduce emissions of environmental pollutants. Another innovation of the model is the multi-cycle modeling of the closed loop of supply chain by considering the uncertainty and the ﬁxed and variable cost of transport. 3. The model formulation In the supply chain, there are a number of i suppliers potentially that supply the raw materials required, and their most suitable ones are chosen to transfer the raw materials to the j producers. There are also options for the construction of recycling centers of k, which by collecting returned products from the collection centers of e, carry out the necessary processing operations aiming to recycling, recovery or disposal of w waste at disposal centers of g and their construction requires an establishment initial cost. In Figure 1, the supply chain is shown. Also, the assumptions regarding the supply chain are as follows: In this supply chain, raw materials are purchased from suppliers and then manufactured in manufacturing plants and delivered to customers. After the expiration of the product’s useful life, a percentage of the goods will be returned to the collection centers and transferred to the recycling centers. Then, depending on the technology used, the recyclable components are removed and after necessary processing they will be re-used in the manufacturing process, and some non-recyclable components are destroyed. From the beginning of the process, all stages are considered deﬁnitive as the materials and components follow a predetermined path. The only return rate is uncertain products that can be due to the useful life of the products and the rate of use. However, it is assumed that the return rate of the products is uncertain. Therefore, modiﬁcations must be made in the model to deal with these uncertain parameters. For lucidity and simplicity, we use the Figure 1. The polyethylene CLSC structure Robust optimization Parameter Description model I Potential suppliers index J Producers index K Recycle index C Costumers index E Index of returned product collecting centers G Index of waste disposal centers L Index of investment options in production centers L’ Index of investment options in recycling centers T Index of time periods (t = 1, [. . .], cl) Ei Environmental impacts index Cl Product life cycle pir Raw material price (r) from supplier (i) psr The price of recycled raw material in the secondary market cjfl The cost of each ﬁnished product production unit (f) using l1 technology in the manufacturers ckfl 0 The cost of each ﬁnished product recycling unit (f) using l2 technology in the manufacturers mfr Number of required raw materials (r) product (f) Fsi The ﬁxed cost of activating the supplier (i) Fmj The ﬁxed cost of the construction of the (j) factory Fee The ﬁxed cost of building a collection center (e) Frk The ﬁxed cost of construction a recycling center (k) Fdg The ﬁxed cost of construction the waste center (g) Utli The cost of using l1 technology in manufacturers Utlr0 The cost of using L2 technology at recycling centers sdfg The average ratio of (f) products sent to the waste unit (g) srfj The average ratio of (f) products sent to the recycling unit (g) dsij Supplier (i) distance from manufacturer (j) dsjc Customer (c) distance from manufacturer (j) dsek Collecting center (e) distance from the recycling center (k) dskj Recycling center (k )distance from producer (j) dskg Recycling center (k )distance from disposal center (g) capiir Maximum capacity of supplier (i) in providing raw material (r) j capjf Producer (j) capacity in production of ﬁnal product (f) caprkf The storage capacity of the recycling center (k) in processing the returned product (f) capmI jmf The storage capacity of the manufacturer (j) for raw materials capj The storage capacity of the manufacturer (j) for ﬁnal materials caprIj The storage capacity of the recycling center (k) for raw materials caprf j The storage capacity of the recycling center (k) for the ﬁnal products vr Weight of a unit of raw material (r) 0 vf Weight of each unit of ﬁnal product (f) 00 vw Weight of each unit of waste (w) Fc The ﬁxed cost of using the vehicle Cvr Variable cost of using the vehicle for per unit of raw material (r) in distance unit 0 Cvf Variable cost of using the vehicle for per unit of ﬁnal product (f) in distance unit 00 Cvw Variable cost of using the vehicle for per unit of waste (w) in distance unit capv Vehicle use capacity 0 Rfce Return rate of ﬁnished product (f) of customer (c) if assigned to collection center (e) Rmrfl 0 Reconstruction rate of the raw material (r) from the returned product (f) in recycling centers in case of using l2 technology Rrrfl 0 Recycling rate of the raw material (r) from the returned product (f) in recycling centers in case of using l2 technology (continued) JM2 Dcft Demand of customer (c) from product (f) during period (t) Hmr The cost of maintaining the raw material (r) in per time unit in the producer in time unit 0 Hmf The cost of maintaining ﬁnished product (f) per time unit in the manufacturer in time unit 0 Hcf The cost of maintaining each returned product (f )in collection centers in time unit Hrr Maintenance cost for per unit of reconstructed raw material (r) at recycling centers per time unit 0 Hrf The cost of maintaining any ﬁnished returned product (f) to recycling centers per time unit drw The cost of disposing each returned waste unit (w) at recycling centers per time unit rff Failure rate of returned product (f) eifpro Environmental impact of production of a product unit (f) sp eiijf Environmental impact of transporting a product unit (f) from supplier (i) to factory (j) pc eijcf Environmental impact of transporting a Product unit (f) from factory (J ) to the customer (c) cn eicef Environmental impact of transporting a unit of product (f) from customer (c) to collection center (e) ni eiekf Environmental impact of transporting a unit of product (f) from the collection center (e) to the recycling center (k) ip eikjf Environmental impact of transporting a unit of product (f) from the recycling center (k) to the factory (j) eijfre Environmental impact of recycling a product unit (f) at the recycling center (k) pr eifg Environmental effect of disposing a unit of product (f) at place (g) subscript f to denote the ﬁnal products and r for the raw materials and w for the waste. Moreover, symbols and notations used through the paper are discussed as follows (i = r, g): The following assumptions are introduced to specify the scope of this study for further model formulation: Assumption 1. There are a number of i potential suppliers in the studied supply chain that prepare the raw materials they need and the most suitable one is chosen for transferring the raw materials to the manufacturers. There are also some other options for construction of recycling centers, which perform the necessary processing operations to k recycle, retrieve or destroy the products through collecting the returned products from the collection centers and their construction requires the initial cost for establishment. Assumption 2. ei presents environmental impacts of green products. For generalization purpose of the model, we do not restrict the environmental impacts of products to a speciﬁc factor. The environmental impact of the network includes the effects that the supply, production, customer, collection and recycling sectors have on the environment, and minimizes the contamination that are from the transportation vehicles. An LCA-based method such as the eco-indicator 99 method has been used to obtain the used coefﬁcients in the equation. Indicator 99 eco-method is used to estimate the environmental impacts of different conﬁgurations in supply chain network. To apply this method, ﬁrstly, the system 0 s limit, its functional unit and the purpose of using the eco-indicator should be deﬁned. Here, the limit of the studied system can be as the boundaries around the supply chain network, which is shown in Figure 1, and the functional unit of the supply chain network can satisfy the effective customer demand by generating and distributing the products in the forwarding network and managing the returned products with a poor quality in an inverse network. The purpose of using the eco-indicator is to estimate the environmental impact of the supply chain network conﬁguration. The product life cycle must be deﬁned in the second step. In our study of polyethylene pipe industry, the life cycle steps include: production pro, transportation from supplier to factory sp, transportation from factory to customer pc; transportation from customer to collection center cn; transportation from collection center to recycling center ni; transportation from recycling centers to factory ip; recycling operation re disposal operation Robust pr. Here, the stage of using in customer centers has been eliminated from the lifecycle steps optimization because it has no effect on the model’s decision-making variables and the overall structure of the supply chain network. model In the third step, materials and processes should be quantiﬁed through the steps of the product life cycle and then, in Step 4, the ﬁnal number is determined by ﬁnding the relevant eco-indicator and multiplying the values in the number obtained by the indicator and gathering the results of the previous step is computed (Goedkoop and Spriensma, 2000). For example, to form a polyethylene tube under the pressure, the amount of pipe material (kg) should be multiplied by the corresponding index, for example, 6.4 (mP/kg), and then, for calculation the ﬁnal peripheral effect of the production stage (pro) raw materials and processes all come together. Other assumptions are expressed as follows: The location of potential facilities is speciﬁed. The return rate of the products is probable. The capacity of facilities’ production and preparation is speciﬁed. The capacity of the vehicles is speciﬁed. All customer’s demand in each period should be met in the base model. The variables of the research are described as follows: The equations for development model of the CLSC model are as follows: X X X X X Minz ¼ Fsi :ysi þ Fmj :ym j þ Fee :yce þ Frk :yrk þ Fdg :ydg e g XXi Xj X X X X kX j i m r r þ Utl :yjl þ Utl 0 :ykl 0 þ cfl :Qjflt Xj X l XX k l0 X X j Xf Xl t X X X þ ckfl 0 :Skfl 0 t þ pir s Xrijt psr Xrk sm k X f X l2 X t i r j t r k t XXX XXX XXX þFcð Nijts þ Njct m þ Nektc þ r Nkjt Þ i j t j c t e k t k j t XXXX XXXX 0 0 s m þ Cvr :dsij :Xrijt þ Cvf :dsjc :Xfjct (1) i X j XX r t j c f t 0 0 X X XX X r r þ Cvf :dsek :Xfekt þ Cvr :dskj :Xrkjt e k f t k j r t X X X X 00 XXX þ Cvw :dskg :wrwkgt þ Hmr :Invm rtj k X g w t j r t 0 0 0 0 X X X X X X XX þ Hmf :Invftjm þ Hcf :Invfte c þ drw :wTwgt Xj X f X t Xe X f X t 0 0 g w t þ Hrr :Invrrtk þ Hrf :Invftk r k r t k f t In the above model, equation (1) shows the ﬁrst objective function. The ﬁrst part of ﬁxed cost of activating the facilities includes the suppliers, manufacturers, recruitment centers, recycling and waste centers, respectively. The next section shows the ﬁxed cost of used technology in production and recycling centers, respectively. The used equipment in this facility is more advanced at the recycling centers and the percentage of recycling and product reconstruction will increase but the cost of launching the equipment increases. Also, JM2 Parameter Variable Description ysi Binary This variable is considered as one if the supplier (i) is chosen, otherwise it will be zero ym j Binary This variable is considered as one if the manufacturer (j) is constructed, otherwise it will be zero yce Binary This variable is considered as one if the collection center of returned products (e) is constructed, otherwise it will be zero yrk Binary This variable is considered as one if the recovery center (k) is constructed, otherwise it will be zero ydg Binary This variable is considered as one if the waste center (g) is embedded, otherwise it will be zero ym jl Binary This variable is considered as one if the production center (j) uses the l1technology, otherwise it will be zero yrkl 0 Binary This variable is considered as one if the recycling center (k) uses the l2 technology, otherwise it will be zero ycce Binary This variable is considered as one if the returned product of customer (c) is assigned to the collection center (e) otherwise it will be zero s Xrijt True variable The amount of raw material (r) delivered from the supplier (i) to the producer (j) at time (t) 0 m Xfjct True variable The amount of ﬁnal material (r) delivered from the producer (j) to the costumer (c) at time (t) 0r Xfekt True variable The amount of ﬁnished returned products (f) delivered from the collection center (e) to the recycling center (k) at time (t) r Xrkjt True variable The amount of recycled material (r) delivered from the recycling center (k) to the manufacturer (j) at time (t) wrwkgt Real variable The amount of waste material (w) transferred from recycling center (k) to waste center (g) at time (t) sm Xrk Real variable The amount of recycled raw material (r) that it is sold in secondary market form the recycling center (k) 0m Invftj Real variable The amount of ﬁnal product (f) that it is stored in storage (j) at the end of period (t) Invm rtj Real variable The amount of raw material (r) that it is stored in manufacturer’s storage (j) at the end of period (t) Invrrtk Real variable The amount of raw material (r) that it is stored in recycling center’s storage (k) at time (t) 0 r Invftk Real variable The amount of returned ﬁnal product (f) that it is stored in recycling center’s storage (k) at the end of period time (t) 0c Invfte Real variable The amount of ﬁnal product (f) that it is stored in collection center’s storage (e) at the end of period (t) wtwgt Real variable The amount of returned waste product (w) that it is disposed in waste center (g) Nijts Integer variable The number of vehicles transferred from supplier (i) to manufacturer (j) m Njct Integer variable The number of vehicles transferred from manufacturer (j) to costumer (c) in period (t) c Nekt Integer variable The number of vehicles transferred from collection center (e) to recycling center (k) in period (t) r Nkjt Integer variable The number of vehicles transferred from recycling center (k) to manufacturer (j) in period (t) d Nkgt Integer variable The number of vehicles transferred from recycling center (k) to waste center (g) in period (t) (continued) Robust Qjﬂt real variable The amount of ﬁnal product (f) produced in manufacturer (j) using technology l1 in period (t) optimization Skfl 0 t Integer The number of returned product (f) processed in recycling center (k) at model time (t) xijr Integer The number of raw material (r) carried from supplier (i) to factory (j) ujkf Integer The number of product (f) carried from factory (j) to costumer (c) yjkf Integer The number of product (f) carried from costumer (c) to collection center (e) vjif Integer The number of product (f) carried from collecting center (e) to recycling center (k) Tjef Integer The number of product (f) carried from collecting center (k) to factory (j) Sgf Integer The number of product (f) disposed in place (g) wi Binary The zero and one variables are the indicators of factory (j) being open or close yj Binary The zero and one variables are the indicators of recycling center (k) being open or close more advanced production facilities reduce the environmental impacts during the production stages. The other part includes production costs and recycling costs. The next section shows the cost of purchasing the raw materials from the suppliers minus the revenue from selling the recycled products, which are non-reusable in production, and are sold in the secondary market. The next section demonstrates the ﬁxed cost of using vehicles, which illustrates the transference from supplier to producer, manufacturer to customer, collection centers to recycling centers and from recycling centers to production centers, respectively. Other sectors show the variable cost of product transferring from the above facilities. Finally, the objective function shows the cost of stock maintenance in producer’s warehouse, collection centers and recycling centers: XXX s Xrijt # M:ysi 8i (2) j r t XXX s Xrijt # M:ym j 8j (3) i r t 0 XXX m Xfjct # M:ym j 8j (4) c f t 0 XXX r Xfekt # M:yce 8e (5) k f t 0 XXX r Xfekt # M:yrk 8k (6) e f t XXX r Xrkjt # M:yrk 8k (7) j r t XXX r JM2 Xrkjt # M:ym j 8j (8) k r t XXX wrwkgt # M:yrk 8k (9) g w t XXX wrwkgt # M:ydg 8g (10) k w t Equation (2) indicates that no primary material is sent from the potential suppliers that are not selected. Equation (3) shows that if no manufacturer factory is constructed, no primary material will be sent to it. Equation (4) shows that a manufacturer’s factory that is not being constructed does not send any product to the customers. Equation (5) ensures that collection centers that are not built do not send any returned products to the recycling centers. Equation (6) indicates that if the recycling center is not established, no returned product from the collection centers will be discharged and due to the equation (7) no recycled material will be sent to the producers. Equation (8) shows that non-constructed production centers do not receive any primary recycled material from the recycling centers. Equation (9) indicates that no waste material will be sent to the waste centers. Equation (10) indicates that waste centers that are not constructed will not receive any waste from waste areas: X ym m jl ¼ yj 8j (11) l1 X ym r jl ¼ yk 8k (12) l Equation (11) shows that manufacturing centers that are activated must be equipped with one of the production technologies. Equation (12) ensures that recycling centers must be equipped with a recycled technology, if constructed and they will pay the ﬁxed cost of using: s X Xrijt : vr Nijts 8i; j; t (13) r cap v 0 0 m m X Xfjct : vf Njct 8j; c; t (14) f capv 0 0 r c X Xfekt : vf Nekt 8e; k; t (15) f capv r r X Xrkjt : vr Nkjt 8k; j; t (16) r capv Equations (13) to (16) calculate the number of vehicles which are sent from the suppliers to the producers, producers to the customers, collection centers to the recycling centers and recycling centers to the producers, in each period: 0 0 X X 0 Invftjm ¼ Invfmðt 1Þj þ Qjflt m Xfjct 8j; f ; t (17) l c 0 Invftjm # capmf j 8j; t (18) Robust X X XX optimization Invrrtk ¼ Invrrðt 1Þk þ s Xrijt þ r Xrkjt mfr :Qjflt 8j; r; t (19) model i k f l X Invm mI rtj # capj 8j; t (20) r XX X Invrrtk ¼ Invrrðt 1Þk þ Rrrfl 0 :mfr : Skfl 0 t r Xrkjt 8k; r; t (21) 0 f j l X Invrrtk # caprIj 8k; t (22) r 0 0 0r X X r Invftk ¼ Invfrðt 1Þk Skfl 0 t þ 1 rff :Xfekt 8k; f ; t (23) l0 e 0 # caprf X r Invftk j 8k; t (24) f 0 0 0 0 X X c Invfte ¼ Invfcðt 1Þe þ Rfce :Dcf ðt c c1 1Þ :yce r Xfekt 8e; f ; t (25) c k Equation (17) calculates the inventory, which is remained in producers’ warehouse in each ﬁnal product at the end of each period (t). This inventory is equal to the remaining inventory from the previous period plus the amount of produced ﬁnal product by the producer in period (t) minus the number of ﬁnal products, which are sent to the customers during that period. Equation (18) indicates that the inventory of producers’ ﬁnal products in each period cannot be greater than their warehouse capacity. Equation (19) shows the inventory of producers’ raw materials in each period. This inventory is equal to the remaining inventory in the previous period plus the amount of raw material received from the suppliers during that period, and the rebuilt raw material from the recycling centers during that period minus the amount of used raw material, which is consumed to produce the ﬁnal products during that period. Equation (20) indicates that the inventory of raw materials stored in the producers’ warehouse in each period cannot exceed the capacity of the raw materials in their warehouse. Equation (21) demonstrates the inventory of raw materials in any warehouses of the recycling centers in each period. This inventory is equal to the inventory of the previous period plus the raw materials that are extracted from the returned products during that period minus the number of raw materials from which the recycling center is sent to the producers. Equation (22) shows the constraint of raw materials 0 warehouse capacity constraints of the recycling centers. Equation (23) indicates the inventory of recycling centers 0 s warehouse from each returned product. This inventory is equal to the remained inventory from the previous period minus the inventory, which is processed during that period plus the inventory sent to that center from the collection centers. Equation (24) shows the capacity constraint of the returned products from the warehouse of each recycling center, which cannot be stored more than the capacity of the warehouse. Equation (25) illustrates the inventory of returned products in the collection centers. This inventory is equal to the remaining inventory from the previous stage plus the amount of returned JM2 products that have expired and returned by the customers minus the number of returned products that are sent to the recycling centers: X ycce ¼ 1 8c (26) e Equation (26) indicates that the returned products from each customer should be assigned to one of the collection centers, but the collection center should have been constructed: ycce # yce 8c; e (27) XX sm Xrk ¼ Rrrfl 0 :mfr :Skfl 0 t 8k; r; t (28) f l0 Equation (27) calculates the number of recycled materials, which are extracted in each recycling centers, but are not reusable in production process and are sold in the secondary market. It is assumed that these products are sold in the same period [Equation (28)]: X 0 m Xfjct ¼ Dcft 8c; f ; t (29) j Equation (29) shows that customer demand for each product in each period should be provided by manufacturers: Qjflt # capjjf :ym jl 8j; f ; l; t (30) 0 Skfl 0 t # capmI r j :ykl 0 8k; f ; l ; t (31) X s Xrijt # capiir 8i; r; t (32) j Equation (30) indicates that the number of produced products cannot exceed the producer’s production capacity. Equation (31) ensures that a recycling center can process the returned products only when it has been activated. The product number cannot exceed its capacity. Equation (32) shows the number of raw materials that can be supplied from each supplier in each period to the capacity of product’s supplies: X X X xijf Sgf ¼ ujkf 8i 2 I; 8g 2 G (33) i g k X X X yjkf Tjef ¼ Sgf 8j 2 J; 8f 2 F (34) e k f X X ujkf 1 s1f s2f xijr ¼ 0 j 2 J; 8f 2 F; 8r 2 R (35) k i X X Tjef s1 f xijr ¼ 0 8j 2 J; 8f 2 F (36) e i X X vjif s2 f xijr ¼ 0 8j 2 J; 8f 2 F (37) i i X X vjif xijr # 0 8i 2 I; 8f 2 F (38) Robust j j optimization Equations (33)-(38) ensures that the environmental impact of the production, transport, model recycling and disposal of products should not exceed the maximum set value: X X X pro sp X X X X pc cn i j r ei f þ ei ijf x ijr þ j ei jcf þ ei cef ujkf yjkf X X c k f X X X ip pr X X X ni re þ i j e k f ei ekf þ ei kjf vjif Tjef þ j g f ei jf þ ei fg # eitmax (39) Equation (39) the ﬁrst part of the function of the second objective indicates the environmental impact of raw materials 0 transportation and production in the factory. The other part demonstrates the environmental impact of ﬁnal product 0 s transportation from the factory to the customer and from the customer to the collection centers. The other part illustrates the environmental impact of transportation of the ﬁnal product from the collection center to the recycling center and recycling center to the factory. The ﬁnal section of equation (39) is about the environmental impact of recycling and disposal of the product: ysi ; ym c r d m r c j ; ye ; yk ; yg ; yjl ; ykl 0 ; yce 2 f0; 1g (40) 0 0 s m r r Xrijt ; Xfjct ; Xfekt ; Xrkjt ; wrwkgt ; Xrk sm 0 (41) 0 0 0 nvftjm ; Invm r r c T rtj ; Invrtk ; Invftk ; Invfte ; wwgt 0 (42) capiir ; capjjf ; caprkf ; capmI mf rI rf j ; capj ; capj ; capj 0 (43) Nijts ; Njct m c ; Nekt r ; Nkjt d ; Nkgt 2N (44) ujkf ; yjkf ; Sgf ; vjif ; Tjef 0 8i 2 I; 8j 2 J; 8e 2 E; 8k 2 K; 8g 2 G; 8f 2 F (45) Equations (40)-(44) specify the type of used variables in the research model. Equation (45) speciﬁes non-negative variables. 3.1 Bertsimas and Sim stable optimization approach Linear programming model is considered as follows in Bertsimas and Sim stable optimization approach (Bertsimas and Sim, 2004): 0 Max C X (46) Subject to: n X Aj xj # b j ¼ 1; . . . ; n (47) j¼1 JM2 X0 (48) If we consider the i-th row of matrix A, Ji is deﬁned as a set of coefﬁcients in row i, which are related to uncertain parameters. Here it is assumed that aij is modeled as a constrained random variable and symmetric with ~a ij that gives a value in the aij ^a ij ; aij þ ^a ij range. aij is deﬁned ð~a ij aij Þ as variable nominal value and ^a ij is its changes radius. Moreover, it is assumed that h ij ¼ ^a ij follows an unknown distribution, which is symmetric gives a value in the (0, 1) range. In this method, a Ci parameter is deﬁned for each row of matrix A, that is not necessarily integer and holds a value in [0,|Ji|] range. CI plays a role as robustness against the conservative system level in the proposed model. It is possible to determine values for non-deterministic parameters in each matrix line in a conservative approach that after the realization of these parameters, the obtained answer from the model is possible with probability 1. Nevertheless, Bertsimas suggests that an answer can be obtained by choosing Ci from the non-deterministic parameters at their critical extreme in each row. Although it cannot be expressed that it remains probable with probability 1, but in return, it remains with an acceptable level of reliability and does not result in excessive deterioration of the target function. Bertsimas proposed the modiﬁed model of linear programming regarding this approach as follows: 0 Max C X (49) Subject to: ( ) X X aij xj þ max ^a ij yi þ ðCi bCi cÞ^a iti yt # bi 8i (50) fsi [fti jsi Ji ;j si j¼bCi c;ti 2 Ji n si j j2si yj # xj # yj (51) L#X #U (52) Y 0 (53) This model is linearized as follows: 0 Max C X (54) Subject to(55) X X aij xj þ Zi Ci þ Pij # bi 8i (56) j j2Ji Zi þ Pij ^a ij yj 8i; j 2 Ji (57) yj # xj # yj (58) L#X #U (59) Y 0 (60) Pij 0 8i; j (61) Robust Zi 0 8i (62) optimization model 3.1.1 Preliminaries. Theorem (1). It is assumed that x* is the optimal solution for the stable optimization problem. Si* and ti* are also considered as set and optimal index in the equation (54), respectively. In this case, the probability that the i-th constraint will be violated is equal to: X X Pr a~ij x*j > b # Pr g ij h ij Ci (63) j j2Ji Parameter g ij in equation (63), is converted to the one in equation (64): 8 > < 1 if j 2 Si* * g ij ¼ a^ij jxj j * and r* ¼ arg min a^ir* jx*r* j (64) : ^ * > if j 2 Ji =Si r2S * [ ft * g air* jxr* j i i and g ij # 1 equation will be established for j 2 Ji nSi* . Theorem (2). If h ij are independent variables for j[Ji and distributed symmetrically in ( 1, 1) range, so: ! X C2i Pr g ij h ij Ci # exp (65) j2J 2jJi j i Theorem (3). a) If h ij are independent variables for j[Ji and distributed symmetrically in ( 1, 1) range, then we have: 8 9 n X 1< n X n = Pr g ij h ij Ci # Bðn; Ci Þ ¼ ð1 m Þ þ (66) 2: bvc l ; j2Ji l¼bvcþ1 To somehow that: Ci þ v n ¼ jJi j; v ¼ ; m ¼v bvc 2 pﬃﬃﬃ b) If Ci ¼ u n is chosen, then we have: ðu 1 y2 lim Bðn; Ci Þ ¼ 1 U ðu Þ; Uðu Þ ¼ pﬃﬃﬃﬃﬃﬃ exp dy (67) n!1 2p 1 2 pﬃﬃﬃ Equation Ci ¼ u n in the previous research studies is most widely used to determine the value of the Ci parameter, which provides the possibility of performing exchanges between optimization and availability. There are indeterminate parameters in the proposed model in equation (25) that should be adjusted using the Bertsimas and Sim model. Consequently, this constraint is rewritten in the form of equations (68) to (72) and equation (25) in the model is replaced by these constraints: 0 0 0 0 c c Z Invfte :C Invfte Inv c þ Invfcðt 1Þe wtwgt þ wtwgt 1 X 0 X fte0 (68) r þ Rfce :Dcft 1 :ycce Xfekt ¼ 0 8e; f ; t c k 0 JM2 0 c Z Invfte ^ :Dcft 1 :yc R 8e; f ; w; c; t (69) fce ce It should be noted that, as the variable y7 (c, s) is always positive and equation (69) is not required. A stable strategy helps a deﬁnitive model to respond to a demand in each scenario and prevents inability in a sample that the facilities0 capacity is enough to respond to requests and returns. 4. Numerical example Modeling was conducted as a multi-cycle modeling in the closed supply chain considering the uncertainty and the ﬁxed and variable costs of transportation in Pars Ethylene Company as a case study. Pars Ethylene Company is a manufacturer of polyethylene pipes and ﬁttings (Figure 2) in various industries such as water, oil and gas, petrochemicals, etc. Pars Ethylene Company is one of the most quality pipe, manifold and polyethylene manhole producers in Iran. It has been involved in most national projects. This cooperation is in the ﬁeld of supply of polyethylene single tube, double-conductor pipe, ﬁttings and polyethylene manhole Ethylene has been for these projects Table 2. The only parameters, which are not shown in Table 3 are the facilities distance and the environmental impacts of production, transportation, recycling and disposal of the products. To identify the distance between the ﬁrst facilities, the facility’s coordinates is developed in an assumed 100 * 100 km area and then the Euclidean distance between the facilities is calculated. In other words, equations (70) and (71) are used for x and y coordinates and then the distance between them is calculated using equation (72): Figure 2. Typical polyethylene products Raw materials and products Amount Cost HDPE 1 Kg 4,670 LDPE 1 Kg 5,290 Table 2. Carbon black 1 Kg 3,820 Quantities of raw Polyethylene pipe 1 Kg 2,100-6,900 materials and HDPE recycled 1 Kg 3,240 products in the LDPE recycled 1 Kg 2,630 sample equation Recycled carbon black 1 Kg 2,400 Parameter Amount Robust optimization Number of potential suppliers Four Number of manufacturers Two model Number of recycling centers Two Number of customers 12 Number of collection centers for returned products Four Number of ﬁnal products Three Number of raw materials Six Number of investment options for recycling centers Three Number of investment options for production centers Four Time periods index Five Product life cycle Seven Raw material (r) price from supplier (i) According to Table 1 Recycled raw material (r) cost in the secondary market According to Table 1 The amount of required raw material (r) from the product (f) According to Table 1 The ﬁxed cost of activating the supplier (i) Uniform (1,000, 2,500) The ﬁxed cost of construction the factory (j) Uniform (800,000, 1,000,000) The ﬁxed cost of construction a collection center (e) Uniform (100,000, 150,000) The ﬁxed cost of construction a recycling center (k) Uniform (150,000, 250,000) The cost of using l1 technology at production centers 50,000 þ 10,000(|l1| l1 þ 1) The cost of using L2 technology at recycling centers 50,000 þ 10,000(|l2| l2 þ 1) The return rate of customer (c) product (0.7, 0.8) Polyethylene pipe (f) sales price (40, 60, 80) Maximum supplier 0 s (i) capacity in providing raw material (r) Uniform (3,000, 8,000) Weight of a unit of raw material (r) One Weight of each unit of ﬁnal product (f) According to the Polyethylene pipe 0 s weight in Table 1 Fixed cost of using the vehicle Uniform (15, 30) Variable cost of using a vehicle in a distance unit Uniform (0.01, 0.025) Vehicle use capacity 1,000 Recovery rate of the raw material (r) from the returned product (f) at the recycling According to Table 3 center (k), in case using l2 technology Raw material recycling rate at the recycling center (k) in case of using l2 technology According to Table 3 The destruction rate of the raw material (r) at the recycling center (k) in the case of Uniform (60,80) using l2 technology The cost of maintaining the raw material (r) per unit time in the manufacturer (i) 0.01*pr1 (i, r) The cost of maintaining the ﬁnal product (f) per unit time in the manufacturer 0.01*uniform (40, 60) The cost of maintaining each returned unit (f) in collecting centers 0.01*uniform (5, 12) Maintenance cost of per unit of raw materials (r) reconstruction at disassembly/ 0.01*pr2 (i, r) recycling centers The maintaining cost of each returned unit (f) in collecting centers 0.01*uniform (6, 10) Production cost using l1 technology Uniform (3.8, 4.6) þ 0.5 (l1 2) Recycle cost using l2 technology Uniform (0.47, 1) þ 0.5 (l2 2) Capacity of the producer 25,000 Capacity of the recycling center 16,000 Supplier’s capacity for delivering the raw materials 200,000 Storage capacity of the collection center Uniform (500, 1,200) Storage capacity of the recycling center Uniform (500, 1,200) Customer demand of polyethylene pipe Uniform (8,400, 180,000) Manufacturer’s capacity (j) for raw materials 20,000 Manufacturer’s capacity (j) for ﬁnal products 5,000 The storage capacity of the recycling center (k) for raw materials 10,000 Table 3. The storage capacity of the recycling center (k) for the ﬁnal products 5,000 Parameter values in Failure rate of the returned product (f) 3% the sample equation JM2 x ¼ uniformð0; 100Þ (70) y ¼ uniformð0; 100Þ (71) qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ distance ð A; BÞ ¼ ðxA xB Þ2 þ ðyA yB Þ2 (72) We will describe how to obtain the indicators in the obtaining method of eco-indicator indexes in environmental objective function after illustrating the parameters of the sample problem in this study Table 4. Obtaining method of eco-indicator indexes in the environmental objective function: Raw materials for polyethylene production including HDPE and LDPE, and carbon black from, pipe production processes and high-voltage power from Attachment are extracted (Goedkoop and Spriensma, 2000). In Table 5, the amount of waste disposal index is derived, and municipal waste index from Attachment are shown (Goedkoop and Spriensma, 2000) Table 7. In Table 6, the values of transportation indicators are derived from attachment. The ﬁrst index indicates the transportation of raw materials from the supplier to the factory, which 16 tons of tracks has considered for this path because the shipments are Recycle (%) Year Table 4. Recycle percentage of 20 2000 (l2 = 3) each investment 60 2006 (l2 = 2) option 90 2010 (l2 = 1) Raw materials and processes Amount Index Total result HDPE 1 Kg 330 330 LDPE 1 Kg 360 360 Carbon black 1 Kg 180 180 Foil’s extrusion blow 1 Kg 2.1 2.1 Table 5. Plastic injection molding 1 Kg 21 21 Production index Forming under pressure 1 Kg 6.4 6.4 (raw materials and High voltage power (high voltage) 1 Kg 22 22 processes) eifpro Total index 1 Kg 921.5 921.5 Index Numerical value Table 6. PE disposal 3.9 Polyethylene pipe 0 s PE municipal waste 1.1 disposal index Total 2.8 usually large in this direction. The second, third, fourth and ﬁfth indexes, are related to the Robust form the factory to the customer centers, product transportation from the customer centers optimization to the collection centers, carrying the product from the collection centers to the recycling sites and transporting the product from the collection centers to disposal centers, model respectively. It is obvious that the amount of transported product is less than 3.5 tons in these paths and this is the reason of choosing the intended index. The index value of Table 8 is derived from attachment (Goedkoop and Spriensma, 2000). The proposed model in addition to the exact solution uses two meta-algorithms including particles swarm algorithm and colonial competition algorithm. The colonial competition algorithm forms the initial set of possible solutions. These initial solutions are known as chromosome in the genetic algorithm, particle in the particles swarm algorithm and country in colonial competition algorithm. Through a particular process, which is set forth below, the colonial competition algorithm gradually improves these initial answers (countries) and eventually provides the suitable answer to the optimization issue (the desired country). The main principles of this algorithm are assimilation policy, imperialistic competition and revolution. This algorithm, by imitating the social, economic and political evolution of countries as well as mathematical modeling of some parts of this process provides such regular operators in an algorithm that can help solve complex optimization problems. In fact, this algorithm scrutinizes the optimization issue in the form of countries and tries to improve the responses during a repetitive process and eventually reach to the optimal answer of the problem. The second algorithm in this study, is the particle swarm algorithm. The algorithm is predicated on the modeling and simulation of birds ﬂying (cluster) or mass movement (grouping) of ﬁsh. Each member of this group is deﬁned by the velocity and position vectors in search space. In each time cycle, the new particles’ position is deﬁned according to the velocity and position vector in the search space. In each time repetition, the new particle position is updated according to the current velocity vector, the best position found by that particle and the best position found by the best particle in the group. 4.1 Genetic algorithm Since 1960, imitation of natural phenomena for use in robust algorithms for solving optimization problems has been called attention to evolutionary computation techniques Numerical value Index sp 34 eiijf pc 140 eijcf Table 7. 140 cn eicef Transportation sp pc ni indexes (eiijf , eijcf , 140 eiekf cn ni ip 140 ip eikjf eicef , eiekf , eikjf ) Index PE recycling Table 8. Polyethylene pipe Numerical value 240 recycling index (eijfre ) JM2 (Holland, 1975). The genetic algorithm ﬁrst proposed by John Holland at the University of Michigan, and the evolutionary strategies and planning developed by Rechenberg, Schwefel, Fogel and Koza, are evolutionary calculation methods. Genetic algorithm is a method of moving from a population of “chromosomes” (e.g. an array of 0 and 1 or multiple “bit”) to another population using a “natural selection” type and genetic algorithm operators such as crossover, mutation and Inversion. Each chromosome is made up of a number of “Gene “(for example, several bits). Each Gene represents a value (for example, 0 or 1). Selection operators select those chromosomes that are permissible to produce, and on average, chromosomes that have a higher degree of conformity produce more children than those with a lower degree of conformity. The intersection operator swaps parts of two chromosomes together to mimic the chromosomes of the reproductive process. The mutation operator changes the values of two parts of one chromosome and the inversion function reverses the order of genes in the chromosome (Mitchell, 1996). The genetic algorithm is summarized in Pseudo Code of GA: t: = 0;// start with an initial time intpopulation p(t);//initialize a usually random population of individuals evaluate p(t);// evaluate ﬁtness of all initial individuals of pop- ulation while ( not down ) do// test for termination criterion ( time, ﬁt- ness, etc.) t: = t þ 1;// increase the time counter p’: = selectparents p(t);// select a sub-population for offspring production recombine p’(t);// recombine the “genes” of selected parents mutate p’(t);// perturb the mated population stochastically evaluatep,p’(t);// evaluate it is new ﬁtness p: = survive p,p’(t);// select the survivors from actual ﬁtness end GA. 4.2 Imperialist competitive algorithm This algorithm was introduced by Atashpaz-Gargari and Lucas (2007). The colonial competition algorithm is inspired by the historical pattern of competition among imperialist countries. In fact, this algorithm is an open door to the world of mathematics with a very human perspective. The algorithm initially starts from several countries in the initial state. In fact, these countries are likely answers to the algorithm. If we want to compare countries with our previous experiences in previous algorithms, countries are the same chromosomes in the genetic algorithm. Thus, countries are divided into two groups, namely, colonial countries and colonial countries. The algorithm slowly improves the states (solutions) with a particular process that is inherent in nature, and ﬁnally, the appropriate solution (optimal state) is obtained. As mentioned, the nature of this algorithm lies in the fundamental properties that constitute the basis for deﬁning this algorithm. The politics of assimilation or assimilation, colonial competition and revolution are important pillars of this algorithm. This algorithm creates algorithms by modeling the social, cultural and economic evolution of countries, and formulates them into mathematical models. This algorithm puts the solutions of the problem, which are the same countries, in a loop of repetition and gradually improves the answers and ﬁnally reaches the optimal solution. The ﬂowchart of the ICA algorithm is shown in Figure 3. As with other algorithms, the algorithm consists of a random population that considers some of the best countries (equivalent to elites in the genetic algorithm) as colonial countries and the rest are Robust optimization model Figure 3. Flowchart of the original ICA algorithm considered as colonial countries (Atashpaz-Gargari and Lucas, 2007). The colonists, by virtue of their power, succeed in attracting countries to themselves. From now on, competition between the colonial countries begins. Any empire that did not survive this competition would be eliminated from the competition and considered itself a colony. The colonial countries also want the colonial countries to advance their survival and survival, which was achieved by the development of civilization and the development of the colony, was accompanied by the relative satisfaction of the people of that country. It is worth noting that public competition was not so important to the colonial countries. Some merely emphasize public competition in the colonial countries with the expansion and transparency of international relations. In fact, behind this friendly and benevolent people, the only policy is to assimilate and assimilate and gain more power and beneﬁts (Atashpaz- Gargari and Lucas, 2007). 4.3 Determine parameters In this section, the proposed method is implemented on a group of Eqs. Taguchi method is used to increase the efﬁciency of the provided algorithms and determine its parameters. Finally, the better algorithm for solving the model is provided according to the statistical methods. Taguchi converts the duplicate data into values, which are a measurement for changes in results. This is an signal to voice ratio (S/N). Section S refers to the desired values and the N part refers to the undesirable ones, which the aim is to maximize this ratio. In other words, JM2 Taguchi recommends a variation analysis using an appropriate S/N that is appropriately selected. Three ratios that are considered as standard ratios are as follows: The best nominal: It is used to reduce the variability of the target value: y2 S=NT ¼ 10log 2 ; (73) s Bigger, Better: It is used if the optimality of the system is achieved when the answer is as large as possible: ! 1 Xn 1 S=NL ¼ 10log i¼1 y2 ; (74) n i Smaller, Better: It is used if the optimality of the system is achieved when the response is as small as possible: 1 Xn 2 S=NS ¼ 10log y i¼1 i ; (75) n N is the number of each test 0 s running and yi is the process response. Given that the objective function is related to lower-cost chain costs, better type; so, the corresponding S/N rate is calculated as follows: 1 Xn 2 S=NS ¼ 10log y i¼1 i ; (76) n In this research, the goal is to ﬁnd the parameters 0 value as the input variable for obtaining the optimal response (Y). Taguchi method for setting the parameters of the proposed genetic algorithm: Taguchi method is used to set the population parameters 0 adjustment (Npop), the probability of intersection (Pc) is the probability of mutation (Pm) and reproduction (Max Gen). In this study, the Taguchi method is used for four factors at three levels, namely, Table 8 shows the parameters 0 values in each level so that the numbers 1, 2 and 3 are the levels of each factor. The numbers in Table 8 are based on the tried and error method. The average values of the parameters for S/N rates at various levels are presented in Figure 4 according to the calculated values for these parameters. Given the equation (76), the goal is to maximize S/N ratio. According to Figure 4, the optimal values of the appropriate parameters are according to the Table 9. Taguchi method for setting the parameters of the proposed colonial competition algorithm: In this section, the goal is to ﬁnd the value of the parameters as the input variable for obtaining the optimal response. The Taguchi method is used at three levels to adjust the parameters of the number of countries (NCountries), the number of imperialisms (NEmp), the number of iterations (Max It) and (PRevolution), which is a probability for revolutions occurrence. Table 10 illustrates the factors0 values in each level, so that the numbers 1, 2 and 3 are the levels of each factor. The numbers in Table 9 are based on a tried and error method Table 11. The average values of the parameters for S/N rates at various levels are presented in Figure 5 according to the calculated values for these parameters. Therefore, the objective function is related to lower-cost chain costs, better type and according to the Figure 5, the optimal values of the appropriate parameters are shown in Table 12. Robust Main Effects Plot for SN ratios optimization Data Means model n pop Pc – 39.5 – 40.0 – 40.5 Mean of SN ratios – 41.0 – 41.5 1 2 3 1 2 3 Pm max Gen – 39.5 – 40.0 – 40.5 – 41.0 Figure 4. The values of the – 41.5 various levels of GA 1 2 3 1 2 3 algorithm parameters Signal-to-noise: Smaller is better in the S/N ratio Parameters One Two Three Pc 0.9 0.85 0.8 Table 9. Pm 0.09 0.12 0.2 Parameters of the Ga npop 150 200 250 algorithm at three Max Gen 100 150 200 levels Parameters Value Pc 0.9 Pm 0.2 Table 10. npop 150 Optimal values for Max Gen 150 Ga parameters In this research, the Lingo 9 software has been used to run the precise model and MATLAB 2013 software has been used to implement the meta-heuristic algorithm on a computer with certain features (CPU corei5 and Ram 8 GB). Other evaluation criteria of those two methods are considered in the following. To study the efﬁciency of the proposed algorithm, the obtained results are compared with the results of the exact method. A relative percentage deviation (RPD) is used, which can be calculated using equation (77) as follows: JM2 ð Algsol Þij Bestsol RPDij ¼ (77) Bestsol where Algsol is the target value (OF) for the given method and Bestsol is the optimal value of the calculated target function through the mathematical model. Obviously, a lower value (RPD) represents a better targeting and better performance of the algorithm, which is less put in local optimization traps. Parameters One Two Three Table 11. NCountries 150 200 250 The parameters of NEmp 4 5 6 Ica algorithm at three Max It 300 400 500 levels PRevolution 0.1 0.15 0.2 Main Effects Plot for SN ratios Data Means N countries N emp – 40.75 – 40.80 – 40.85 Mean of SN ratios – 40.90 – 40.95 1 2 3 1 2 3 max it P revolution – 40.75 – 40.80 – 40.85 Figure 5. – 40.90 The values of the – 40.95 various levels of ICA 1 2 3 1 2 3 algorithm parameters in the S/N ratio Signal-to-noise: Smaller is better Parameters Value NCountries 200 Table 12. NEmp 5 Optimal values for Max It 300 Ica parameters PRevolution 0.2 Eq value Mathematical model Genetic algorithm Colonial competition algorithm Eq number Dimensions Optimum Time average best time RPD average best time RPD 1 Five 30.4261 00:12:12 30.4261 30.4261 00:00:3.5124 0 30.4261 30.4261 00:00:0.8182 0 2 Five 19.0966 00:40:23 19.0996 19.0996 00:00:3.207 0 19.0996 19.0996 00:00:0.9152 0 3 10 – – 45.397 45.3793 00:00:17.621 0.0019 46.0974 46.0192 00:00:2.0852 0.0085 4 10 – – 34.7814 34.7814 00:00:15.543 0.037 35.4319 35.2014 00:00:1.802 0.0327 5 15 – – 66.565 65.2738 00:00:32.341 0.0989 67.6563 67.0128 00:00:2.2237 0.048 6 15 – – 46.7088 45.6543 00:00:23.937 0.1154 48.4848 46.3034 00:00:3.009 0.1059 7 30 – – 55.99 54.4008 00:00:50.486 0.1464 58.7096 57.4328 00:00:6.1276 0.1111 8 30 – – 47.2664 45.8593 00:00:47.385 0.1534 47.5821 46.2892 00:00:6.4917 0.1396 9 35 – – 65.632 62.3712 00:00:60.5741 0.2614 66.5495 64.9282 00:00:7.4184 0.1248 10 35 – – 52.352 50.4288 00:00:56.361 0.209 51.7797 50.4288 00:00:7.5395 0.134 11 40 – – 76.1501 72.1927 00:00:60.953 0.274 74.3602 73.353 00:00:8.8584 0.1231 12 40 – – 59.8185 57.601 00:00:59.648 0.1924 57.386 55.2714 00:00:8.0041 0.1912 13 50 – – 72.9573 69.932 00:00:71.914 0.2164 73.2482 71.0941 00:00:11.9206 0.1515 14 50 – – 50.289 46.7211 00:00:57.726 0.3539 48.9053 47.2771 00:00:11.3543 0.1721 15 60 – – 86.3173 82.1804 00:00:69.215 0.2516 84.0378 81.603 00:00:13.4603 0.1493 16 60 – – 56.5441 52.6678 00:00:65.176 0.368 55.054 53.1258 00:00:13.1975 0.1814 17 70 – – 64.0549 61.4331 00:00:83.487 0.2133 63.6018 61.5832 00:00:13.5725 0.1638 18 70 – – 54.0233 50.6812 00:00:88.595 0.3297 53.4794 51.490 00:00:13.3581 0.1931 19 100 – – 88.2091 84.447 00:00:102.673 0.4227 86.9012 84.2476 00:00:20.0561 0.1574 20 100 – – 77.5891 70.8454 00:00:92.012 0.4759 73.01 70.2116 00:00:19.306 0.1992 The obtained results colonial competition optimization method, genetic from the exact algorithm and Table 13. Robust model JM2 It is necessary to note that for the times that the mathematical model needs more than an hour to reach the optimal solution, the solution is an local optimal, otherwise the best given answer is used as Bestsol. Genetic algorithm and proposed colonial competition have been run for 20 problems with random data generated from the described method in the previous section and each problem is implemented ﬁve times. The obtained results from the implementation of the exact method and the meta- algorithms for sample problems are shown in Table 13. Table 13 shows the obtained results after solving the model in different dimensions. The results show that the results of the genetic algorithm for the best index have better quality than the colonial competition algorithm. However, with increasing problem dimensions, the results for the mean index show that the results of the colonial competition algorithm have better quality than the genetic algorithm. On the other hand, the computational time associated with the exact algorithm increases regarding the samples 0 size growing. As it is demonstrated S in the table, there is also a size of 5 3 that precision algorithm achieves the optimal response at the time of 00:40:23, while for exact size of 2 10, it does not achieve any local or global optimal answer. The results of colonial competition algorithm are investigated in the following. As it is mentioned, the RPD criterion is used to evaluate the performance of the proposed meta-heuristic algorithms, which the presented RPD in this table is equal to the calculated average RPD for the ﬁve-time execution of each Eq. To see the process of changing the RPD criterion for changes in the dimensions of the problem, the diagram is shown in Figure 6. As it is clear from this chart, both RPD values are equal to zero for small problems, which indicates the correct functioning of the genetic algorithm and the proposed colonial competition. In larger-dimensional problems that the precise method cannot solve a problem at a reasonable time, the RPD value is calculated in RPD 0.8 0.6 0.4 0.2 Figure 6. RPD diagram for 0 genetic algorithm and 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 colonial competition ICA GA TIME 200 150 100 Figure 7. 50 Computational time graph of genetic 0 algorithm and 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 colonial competition ICA GA terms of the best solution obtained from the genetic algorithm and colonial competition Robust in ﬁve replicates. In Figure 6, it is shown that, the RPD value in the colonial competition optimization algorithm has a gentler gradient than the genetic algorithm, which indicates a better performance of this algorithm than the genetic algorithm. model Another criterion for evaluating the performance of proposed algorithms is the time to solve problems by algorithms, which provides a graph of the computational time of the genetic algorithm and the proposed colonial competition for different problem sizes in Figure 7. As it is shown in Figure 7, the computational time that is associated with small- dimensional problems is very small comparing the exact solution. The computational time of the genetic algorithm is signiﬁcantly more than the colonial competition algorithm and has a gentler gradient than the colonial competition algorithm. Now, a deﬁnite model without taking environmental constraints into account is solved. Then, changes in problems solution due to activating the environmental constraints are demonstrated. Finally, the effect of uncertainty in the problem 0 s solution is investigated using the optimization method of Bertsimas and Sim. Production costs, transportation cost, ﬁxed costs of establishing the production and distribution centers, and the cost of penalty for unused capacity at production and distribution centers after solving the deﬁnitive models are given in Table 14. 4.4 Deﬁnite model without environmental constraints Deﬁnitive model is considered with average of uncertain parameters, environmental constraints do not consider as active and environmental label of products and cost models are calculated. So, the results are described in Table 15. It is noted that choosing advanced technology in the recycling centers is economical even without activating the environmental constraints. This is because of the savings that are made in purchasing raw materials and sailing the recycled products in secondary markets. However, using the advanced technology is not feasible in cost centers because despite the reduction in environmental impacts, there is no reason to use it without activating environmental constraints due to the investment costs. Total cost 63,352,674 The cost of facilities construction 3,906,129 Transportation cost 16,854,380 Production cost 6,233,020 Recycling cost 588,921 The cost of purchasing raw materials 31,539,844 The income from sailing recycled products 2,801,726.5 Maintenance cost 1,088,653.3 The cost of environmental investment 340,000 Active suppliers 1, 2, 3 Table 14. Active producers 1, 2, 3 Conclusive model Active collecting centers 2 Active recycling centers 1, 2 results without Used technology in production centers Technology 3 taking environmental Used technology in recycling centers Technology 1 constraints into Costumers’ collecting centers Collection center 2 consideration JM2 Total cost 66,683,130.29 The cost of facilities’s Construction 39,69,526 Transportation cost 18,525,560.6 Production cost 6,836,174 Recycling cost 624,854 The cost of purchasing raw materials 32,038,819.22 The income from sailing recycled products 3,101,667.1 Maintenance cost 1,126,529.37 The cost of environmental investment 460,000 Table 15. Active suppliers 1, 2, 3 Active producers 1, 2, 3 The results of Active collecting centers 2, 3 deﬁnitive model Active recycling centers 1, 2, 3 considering the Used technology in production centers Technology 3 environmental Used technology in recycling centers Technology 1 constraints Costumers’ collecting centers Costumers 1 and 2 to Center 1 and others to Center 2 4.5 A deﬁnitive model considering the environmental constraints All changes in the results due to activating environmental constraints are illustrated in the next stage. The results of the deﬁnitive model by considering the environmental constraints are described in Table 15. In this case, it is noted that the total cost of the previous mode has increased by 330,455.99 units, but the used technology in production and recycling centers has not changed. So, the transport distance between the facilities is reduced because of the construction of more collection and recycling centers and more products are recycled, which reduces the environmental impact and environmental constraints. 4.6 Uncertain model results It should be noted that the results of the solved models using the robust optimization approach are as probable as the results of the model considering the realization of different values for the uncertain parameters and the degree of objective function corresponding to the level of reliability of the model. This means that if we apply a completely conservative approach to ﬁnd an answer that is non-deterministic for any parameter; this may result in a target function that does not have the desired value for decision-makers. Therefore, if the results are presented in a way that demonstrates the probability of the model and the value The percentage of increasing the The value of U (u ) probability target function vs moderate level target function u of the model – 63,394,230 0 50 0.3 60,582,269 0.25 60 0.6 60,763,951 0.52 70 0.9 60,942,860 0.84 80 Table 16. 1.2 61,110,641 1.28 90 The value of target 1.4 61,229,392 1.64 95 function for different 1.6 61,332,516 1.96 97.5 values of the model’s 1.8 61,423,076 2.33 99 conﬁdence level 1.8 61,423,076 – 100 of the corresponding target function, the decision-makers can easily choose the appropriate Robust answer given their risk-taking. The various results of the model feasibility and model optimization optimization are presented in the Table 16. It is clear that when the level of model’s assurance increases, the value of the objective model function increases and the value of the objective function becomes worse. This is a good result because increasing the objective function means the cost of accepting less risk. These results are very valuable to decision-makers of the organization because they choose the solution, given their risk level, which is more desirable for them. The changes in the objective function are shown in relation to the level of functionality of the model in the graph of Figure 8. Figure 9 is also taken from the results of Table 16 and it shows that the value of the objective function increases and increasing the level of reliability of the model. However, in this research, we are trying to minimize the value of the target pan because the target function is costly and the best value of the target function is necessarily the least possible. This conclusion is true because increasing the objective function means that we are less inclined to accept risk. This increase in the value of the objective function is due to the level of risk-taking of the managers and indicates that the costs will inevitably increase to reduce the existing risks. 4.7 Sensitivity analysis Sensitivity analysis in the proposed issue, raw material prices, the cost of investment in technology and customer demand are known as parameters that can ﬂuctuate in the real world. Sensitivity analysis in performed at ﬁve levels, in this research. These levels include 20% and 40% increasing, and 20% and 40% decreasing in the considered parameter. The Total Cost considering the environmental limitaƟons without environmental limitaƟons 7 5 3 Figure 8. Compare costs in 1 both cases 0 20,000,000 40,000,000 60,000,000 80,000,000 61,500,000 The amount of target funcƟon 61,400,000 99,100 61,300,000 97.5 61,200,000 95 61,100,000 90 61,000,000 80 60,900,000 60,800,000 70 Figure 9. 60,700,000 The value of the 60,600,000 60 objective function vs 60,500,000 the probability of the 0 20 40 60 80 100 120 model Probability of the steady model's soluƟon JM2 results of changing in parameters are shown in Table 17. þ and signs in this table demonstrate the increase or the reduction to the desired level, respectively. It should be noted that every time the model is run because of sensitivity analysis, all values are equal to their initial values and only the considered parameter is changed. Regarding the price of raw materials, considering that the highest level of technology is used to provide the environmental constraints in an optimal solution, a linear relationship between its increase and decrease with the cost of the model is expected. As it was expected, costs increase with increasing the demand, and the reason is the ﬁxed cost of supplying raw materials, increasing shipping costs, etc. However, as it is shown in Table 17, more than 40% increase in demand will cause the model to fail and some constraints such as capacity or even environmental ones are violated. 5. Conclusion Given the ﬁnancial bottlenecks facing businesses in Iran and the pressures on the environmental aspects of production and distribution, it is vital to consider these issues in designing different dimensions of business. In addition to minimizing the total cost, environmental impacts of production, transportation and distribution of products were also minimized. Bertsimas and Sim method are used to determine the uncertainty about the rate of return products. Investigation of the results of the solved model shows the positive effect of the assurance level of the model on the objective function value. Overall, the results show that increased costs are because of lower risk acceptance. Compliance with environmental requirements also has a direct impact on total costs. From the practical point of view, it is suggested that strategic supply chain level decisions be integrated into line with environmental decisions. Such an approach not only reduces the risk in production and distribution but also improves supply chain performance both in cost and environmental terms. Also, the positive effect of observing uncertainty in the model indicates that manufacturers can improve the quality of their decisions in terms of production and distribution and design of the supply chain by considering the amount of returned goods. In this study, to avoid the subtleties optimization of the network, the forward and reverse model are considered as one, and to satisfy the environmental facet of the chain, a lifecycle- based approach called the eco-indicator 99 is used. The proposed model is implemented in the polyethylene pipe industry and the results of the research model implementation are described. Given the results, the results of the proposed colonial competition algorithm have the higher quality than the genetic algorithm, which includes RPD and computational time of the algorithm, and is signiﬁcantly less than the genetic algorithm. Therefore, the proposed algorithm of this research is the colonial competition algorithm, based on the obtained results. Also, the constraint test and the sensitivity analysis results demonstrate that the proposed model in this research is sufﬁciently accurate. Moreover, the results reveal that the consideration of environmental constraints would impose additional costs on the supply chain anyway. This extra cost can be justiﬁed looking Table 17. Changes in the (%) objective function vs parameters change %40þ %20þ 0 %20– %40– changes in raw material cost Raw material’s price 74,249,721 67,746,836 60,394,230 54,700,038 47,885,788 parameters and Technology’s cost 61,421,387 61,325,403 60,394,230 61,133,381 61,037,395 products demand Products’ demand – 74,982,225 60,394,230 47,353,334 35,341,422 at social responsibility towards the environment and human destiny. Yet, in many Robust developed countries because of the advancement of technology, economic efﬁciency is optimization considered as much as environmental efﬁcacy. In the present study, it is shown that the technology used in recycling centers with savings at the cost of raw material supply not model only does justify the environmental issues but also has economic justiﬁcation. 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(2009), “A stochastic optimization model for integrated forward/ reverse logistics network design”, Journal of Manufacturing Systems, Vol. 28 No. 4, pp. 107-114. Taleizadeh, A.A., Karimi Mamaghan, M. and Torabi, S.A. (2018), “A possibilistic closed-loop supply chain: pricing, advertising and remanufacturing optimization”, Neural Computing and Applications, Vol. 32 No. 4, doi: 10.1007/s00521-018-3646-3. Corresponding author Ehsan Sadeh can be contacted at:

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Appendix Robust optimization model Indicator Description ABC 400 Three HDPE 330 One LDPE 360 One PA 6.6 630 Three PC 510 One PET 380 Three PET bottle grade 390 Used for bottles Three PP 330 Three PS (GPPS) 370 General purposes Three PS (HIOS) 360 High impact One PS (EPS) 360 Expandable Three PUR energy absorbing 490 Three PUR ﬂexible block foam 480 For furniture, bedding, clothing Three PUR hardfoam 420 Used in white goods, insulation, construction One material PUR semi rigid foam 480 Three PVC high impact 280 Without metal stabilizer (Pb or Ba) and without One plasticizer (see under chemicals) PVC (rigid) 270 Rigid PVC with 10% plasticizers (crude One estimate) PVC (ﬂexible) 240 Flexible PVC with 50% plasticizers (crude One estimate) PVDC 440 For thin coatings Three Processing of plastics (in millipoints) Blow foil extrusion PE 2.1 Per kg PE granulate, but without Two production of PE. Foil to be used for bags Calandering PVC foil 3.7 Per kg PVC granulate, but without Two production of PVC Injection Moulding – 1 21 Per kg PE, PP, PS, ABS, without Four production of material Injection Moulding – 2 44 Per kg PVC, PC, without production of Four material Milling, turning, crilling 6.4 Per dm3 machined material, without Four production of lost material Pressure forming 6.4 Per kg Four React. Inj. Moulding – PUR 12 Per kg, without production of PUR and Four possible other components Ultrasonic welding 0.098 Per m welded length Four Vacuum – forming 9.1 Per kg material, but without production Four of material Polythene landﬁll indicators Landﬁll Controlled landﬁll site. 78% of municipal waste in Europe is landﬁlled Table A1. Landﬁll PE 3.9 Two Indices of raw Landﬁll PP 3.5 Two material production (continued) of polyethylene JM2 Landﬁll PET 3.1 Two Landﬁll PS 4.1 Indicator can also be used for landﬁll of Two ABS Landﬁll EPS foam 7.4 PS foam, 40 kg/m3, large volume Two Landﬁll foam 20 kg/m3 9.7 Landﬁll of foam like PUR with 20 kg/ Two m3 Landﬁll foam 100 kg/m3 4.3 Landﬁll of foam like PUR with 100 kg/ Two m3 Landﬁll Nylon 3.6 Two Landﬁll PVC 2.8 Excluding leaching of metal stabilizer Two Indicators of urban waste polyethylene materials Municipal waste In Europe, 22% of municipal waste is incinerated, 78% is landﬁlled Indicator is not valid for voluminous waste and secondary materials Municipal waste PE 1.1 Two Municipal waste PP 0.13 Two Municipal waste PET 1 Two Municipal waste PS 2 Not valid for foam products Two Municipal waste Nylon 3.1 Two Municipal waste PVC 10 Two Municipal waste PVDC 16 Two Municipal waste Paper 0.71 Two Municipal waste Cardboard 0.64 Two Municipal waste ECCS steel 5.9 Valid for primary steel only! Two Municipal waste Aluminium 23 Valid for primary Aluminium only! Two Municipal waste Glass 2.2 Two Transport (in millipoints per tkm) Including fuel production Delivery van <3.5t 140 Road transport with 30% load, 33% petrol unleaded, One 38% petrol leaded, 29% diesel (38% without catalyst) (European average including return) Truck 16t 34 Road transport with 40% load, (European average One including return) Truck 28t 22 Road transport with 40% load, (European average One including return) Truck 28t (volume) 5 Road transport per m3km. use when volume instead One of load is limiting factor Truck 40t 15 Road transport with 50% load, (European average One including return) Passenger car W- Europe 29 Road transport per km One Rail transport 39 Rail transport, 20% diesel and 80% electric traind One Tanker inland 5 Water transport with 65% load (European average One including return) Tanker oceanic 0.8 Water transport with 54% load (European average One including return) Freighter inland 5.1 Water transport with 70% load (European average One including return) Table A1. (continued) Robust Freighter oceanic 1.1 Water transport with 70% load (European average One optimization including return) model Average air transport 78 Air transport with 78% load (Average of all ﬂights) Six Continental air transport 120 Air transport in a Boeing 737 with 62% load Six (Average of all ﬂights) Intercontinental air transport 80 Air transport in a Boeing 747 with 78% load Six (Average of all ﬂights) Intercontinental air transport 72 Air transport in a Boeing 767 or MD11 with 71% load Six (Average of all ﬂights) Table A1. Attachment: Tables eco-indicator 99 The tables that have been extracted from the corresponding eco-indicators of the eco-indicators are given below (Goedkoop and Spriensma, 2000). For instructions on how to order reprints of this article, please visit our website: www.emeraldgrouppublishing.com/licensing/reprints.htm Or contact us for further details:

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